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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1omo | Structured version Visualization version GIF version |
Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 48690 assuming ax-un 7754 (see f1omoALT 48692). (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
f1omo.1 | ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) |
Ref | Expression |
---|---|
f1omo | ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8515 | . . . 4 ⊢ 1o ∈ V | |
2 | eqid 2735 | . . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋) | |
3 | 1, 2 | fvconst0ci 48689 | . . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) |
4 | mo0 48662 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) | |
5 | el1o 8532 | . . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
6 | el1o 8532 | . . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
7 | eqtr3 2761 | . . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥) | |
8 | 5, 6, 7 | syl2anb 598 | . . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) |
9 | 8 | gen2 1793 | . . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) |
10 | eleq1w 2822 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o)) | |
11 | 10 | mo4 2564 | . . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)) |
12 | 9, 11 | mpbir 231 | . . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o |
13 | eleq2w2 2731 | . . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o)) | |
14 | 13 | mobidv 2547 | . . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o)) |
15 | 12, 14 | mpbiri 258 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) |
16 | 4, 15 | jaoi 857 | . . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) |
17 | 3, 16 | ax-mp 5 | . 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) |
18 | f1omo.1 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) | |
19 | 18 | fveq1d 6909 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) |
20 | 19 | eleq2d 2825 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) |
21 | 20 | mobidv 2547 | . 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) |
22 | 17, 21 | mpbiri 258 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 ∅c0 4339 {csn 4631 × cxp 5687 ‘cfv 6563 1oc1o 8498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1o 8505 |
This theorem is referenced by: indthinc 48853 prsthinc 48855 |
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